The Self-Consistent Expansion: From Divergent Perturbation Theory To Exponential Convergence

  Benjamin Remez  ,  Moshe Goldstein  
Tel Aviv University

For many nonlinear physical systems, an approximate solution is pursued by conventional perturbation theory (PT) in powers of the non-linear term. However, this frequently produces a divergent asymptotic expansion which does not permit high-accuracy solutions, particularly for strong couplings.

An alternative method, the Self-Consistent Expansion (SCE), has been introduced by Schwartz and Edwards. Its basic idea is a rescaling of the zeroth system around which the solution is expanded, to achieve optimal results. It can be seen as a procedure to systematically improve upon the variational approximation.

While low-order SCE calculations have been remarkably successful in describing the dynamics of nonequilibirium many-body systems (e.g., the Kardar-Parisi-Zhang equation), its convergence properties have not been elucidated before. To address this issue we apply this technique to the canonical partition function of the classical harmonic oscillator with a quartic gx4 anharmonicity, for which PT's divergence is well-known. We explicitly obtain the nth order SCE approximation for the partition function, which is rigorously found to converge exponentially fast in n, and uniformly in g, for any coupling g>0. Thus,

SCE permits the calculation of the partition function to arbitrary precision, even in the strong-coupling regime g>>1. SCE is shown to carry over to the many-particle and quantum cases.